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The Integer Nucleolus of Directed Simple Games: A Characterization and an Algorithm

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  • Reiner Wolff

    (Department of Economics, University of Fribourg, 1700 Fribourg, Switzerland)

Abstract

We study the class of directed simple games, assuming that only integer solutions are admitted; i.e., the players share a resource that comes in discrete units. We show that the integer nucleolus—if nonempty—of such a game is composed of the images of a particular payoff vector under all symmetries of the game. This payoff vector belongs to the set of integer imputations that weakly preserve the desirability relation between the players. We propose an algorithm for finding the integer nucleolus of any directed simple game with a nonempty integer imputation set. The algorithm supports the parallel execution of multiple threads in a computer application. We also consider the integer prenucleolus and the class of directed generalized simple games.

Suggested Citation

  • Reiner Wolff, 2017. "The Integer Nucleolus of Directed Simple Games: A Characterization and an Algorithm," Games, MDPI, vol. 8(1), pages 1-12, February.
    • Handle: RePEc:gam:jgames:v:8:y:2017:i:1:p:16-:d:91763
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    References listed on IDEAS

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    3. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Carreras, Francesc & Freixas, Josep, 1996. "Complete simple games," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 139-155, October.
    5. Kurz, Sascha & Napel, Stefan & Nohn, Andreas, 2014. "The nucleolus of large majority games," Economics Letters, Elsevier, vol. 123(2), pages 139-143.
    6. Reiner Wolff & Yavuz Karagök, 2012. "Consistent allocation of cabinet seats: the Swiss Magic Formula," Public Choice, Springer, vol. 150(3), pages 547-559, March.
    7. Maschler, Michael, 1992. "The bargaining set, kernel, and nucleolus," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 18, pages 591-667, Elsevier.
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    More about this item

    Keywords

    integer nucleolus; integer prenucleolus; desirability relation; simple games; JEL Classification; C63; C71; D72;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior

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